A system with coordinates $q_{i}, i=1, \ldots, n$, has the Lagrangian $L\left(q_{i}, \dot{q}_{i}\right)$. Define the energy $E$.

Consider a charged particle, of mass $m$ and charge $e$, moving with velocity $\mathbf{v}$ in the presence of a magnetic field $\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{A}$. The usual vector equation of motion can be derived from the Lagrangian

$$L=\frac{1}{2} m \mathbf{v}^{2}+e \mathbf{v} \cdot \mathbf{A}$$

where $\mathbf{A}$ is the vector potential.

The particle moves in the presence of a field such that

$$\mathbf{A}=(0, r g(z), 0), \quad g(z)>0$$

referred to cylindrical polar coordinates $(r, \phi, z)$. Obtain two constants of the motion, and write down the Lagrangian equations of motion obtained by variation of $r, \phi$ and $z$.

Show that, if the particle is projected from the point $\left(r_{0}, \phi_{0}, z_{0}\right)$ with velocity $\left(0,-2(e / m) r_{0} g\left(z_{0}\right), 0\right)$, it will describe a circular orbit provided that $g^{\prime}\left(z_{0}\right)=0$.